hey Wicked, glad you're finding the fun in DF.
Unless things have changed since I've played(less than a year), this is the trading process:
1) Caravan arrives
2) In the 'q' of trade depot you select which items of yours to bring to trade depot.
3) Via 'q' of trade depot you say someone is needed to go barter with caravan.
4) Again via 'q' of trade depot you enter the trading screen in which you select the items you are offering from your stock for the selected items of their stock.
5) If trade is accepted those items switch lists(yours become theirs, theirs become yours), and if rejected you can adjust what you are offering/asking for. Sometimes there is a counter-offer already selected, the dialog will indicate if this is the case.
6) When trading is done you say the bartering dwarf is no longer required.
7) Items are put away.
In 2, beware that you can select entire categories. You can also shift-enter and select all the individual items listed on the screen at that moment. And if you select some, then decide you just want to take them all and shift-enter those first ones will be unselected. (Shift-enter really just hits enter on each and every line on screen, flipping their status.)
In 3 you can restrict the bartering to the broker, which is a 'n'oble position. If you do this but don't have a broker assigned, no one will come. (I'm not even sure you can do that.)
The values listed next to items will be more more or less estimates depending on your bartering dwarf's skill. Keep in mind there is also a weight associated with each item and their wagon has a weight limit. This most likely won't matter, but I did once have a trade offer rejected because I wanted so many light-weight items from them and was offering a bunch of stone in return.
Each trade exchange is used to improve the dwarf's skill; so if you want you can break your entire exchange into subsets and get more experience. Just remember that the items immediately swap lists, so the stuff you gave them will now populate(clutter, really) their list and vice-versa; don't trade away something you just bought from them!
After bartering is done, everything you own in the trade depot--stuff you just bought and stuff you brought to trade away but didn't--is your stuff and isn't in a stockpile. So jobs will be generated to take those things and put them where they belong. Of course that won't happen if you don't have a stockpile for that type of item or that stockpile is full. Also it won't happen if the input to that stockpile is limited by linking.
From my personal experience, I think the job to put stuff away is pretty low priority--stuff has sat there for ages. And now that I'm writing this I'm not actually sure I've seen a bunch of unassigned hauling tasks in 'j'ob list after bartering has completed, so maybe there is a problem with that process?
Workaround would probably be to dump everything in the depot and then reclaim them.